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Digital Logic Circuits, Digital
Component and Data
Representation
Course: MCA-I
Subject: Computer Organization
And Architecture
Unit-11
Basic Definitions
• Binary Operators
– AND
z = x • y = x y z=1 if x=1 AND y=1
– OR
z = x + y z=1 if x=1 OR y=1
– NOT
z = x = x’ z=1 if x=0
• Boolean Algebra
– Binary Variables: only ‘0’ and ‘1’ values
– Algebraic Manipulation
Boolean Algebra Postulates
• Commutative Law
x • y = y • x x + y = y + x
• Identity Element
x • 1 = x x + 0 = x
• Complement
x • x’ = 0 x + x’ = 1
Boolean Algebra Theorems• Duality
– The dual of a Boolean algebraic expression is obtained by interchanging the AND and the OR operators and replacing the 1’s by 0’s and the 0’s by 1’s.
– x • ( y + z ) = ( x • y ) + ( x • z )
– x + ( y • z ) = ( x + y ) • ( x + z )
• Theorem 1
– x • x = x x + x = x
• Theorem 2
– x • 0 = 0 x + 1 = 1
Applied to a
valid equation
produces a
valid equation
Boolean Algebra Theorems
• Theorem 3: Involution– ( x’ )’ = x ( x ) = x
• Theorem 4: Associative & Distributive– ( x • y ) • z = x • ( y • z ) ( x + y ) + z = x + ( y + z )
– x • ( y + z ) = ( x • y ) + ( x • z )
x + ( y • z ) = ( x + y ) • ( x + z )
• Theorem 5: De Morgan– ( x • y )’ = x’ + y’ ( x + y )’ = x’ • y’
– ( x • y ) = x + y ( x + y ) = x • y
• Theorem 6: Absorption– x • ( x + y ) = x x + ( x • y ) = x
Boolean Functions
• Boolean Expression
Example: F = x + y’ z
• Truth Table
All possible combinationsof input variables
• Logic Circuit
x y z F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 1
1 1 1 1x
yz
F
Algebraic Manipulation
• Literal:
A single variable within a term that may be complemented or not.
• Use Boolean Algebra to simplify Boolean functions to produce simpler circuits
Example: Simplify to a minimum number of literals
F = x + x’ y ( 3 Literals)
= x + ( x’ y )
= ( x + x’ ) ( x + y )
= ( 1 ) ( x + y ) = x + y ( 2 Literals)Distributive law (+ over
•)
Complement of a Function
• DeMorgan’s Theorm
• Duality & Literal Complement
CBAF
CBAF
CBAF
CBAF
CBAF
CBAF
Canonical Forms
• Minterm
– Product (AND function)
– Contains all variables
– Evaluates to ‘1’ for aspecific combination
Example
A = 0 A B C
B = 0 (0) • (0) • (0)
C = 0
1 • 1 • 1 = 1
A B C Minterm
0 0 0 0 m0
1 0 0 1 m1
2 0 1 0 m2
3 0 1 1 m3
4 1 0 0 m4
5 1 0 1 m5
6 1 1 0 m6
7 1 1 1 m7
Canonical Forms
• Maxterm
– Sum (OR function)
– Contains all variables
– Evaluates to ‘0’ for a
specific combination
Example
A = 1 A B C
B = 1 (1) + (1) + (1)
C = 10 + 0 + 0 = 0
A B C Maxterm
0 0 0 0 M0
1 0 0 1 M1
2 0 1 0 M2
3 0 1 1 M3
4 1 0 0 M4
5 1 0 1 M5
6 1 1 0 M6
7 1 1 1 M7
Canonical Forms
• Truth Table to Boolean FunctionCBAF CBA CBA ABCA B C F
0 0 0 0
0 0 1 1
0 1 0 0
0 1 1 0
1 0 0 1
1 0 1 1
1 1 0 0
1 1 1 1
Canonical Forms
• Sum of Minterms
• Product of Maxterms
ABCCBACBACBAF
7541 mmmmF
)7,5,4,1(F
CABBCACBACBAF
CABBCACBACBAF
CABBCACBACBAF
))()()(( CBACBACBACBAF
6320 MMMMF
(0,2,3,6)F
Standard Forms• Sum of Products (SOP)
ABCCBACBACBAF BA
BA
CCBA
)1(
)(
AC
BBAC
)(
CB
AACB
)(
)()()( BBACCCBAAACBF
ACBACBF
Standard Forms• Product of Sums (POS)
CABBCACBACBAF
)( CCBA
)( AACB
)( BBCA
)()()( AACBCCBABBCAF
CBBACAF
))()(( CBBACAF
Two-Level Implementations• Sum of Products (SOP)
• Product of Sums (POS)
B’C
FB’A
AC
AC
FB’A
B’C
ACBACBF
))()(( CBBACAF
Logic Operators
• AND
• NAND (Not AND)
xy
x • y
xy
x • y
x y AND
0 0 0
0 1 0
1 0 0
1 1 1
x y NAND
0 0 1
0 1 1
1 0 1
1 1 0
Logic Operators
• OR
• NOR (Not OR)
xy
x + y
xy
x + y
x y OR
0 0 0
0 1 1
1 0 1
1 1 1
x y NOR
0 0 1
0 1 0
1 0 0
1 1 0
Logic Operators
• XOR (Exclusive-OR)
• XNOR (Exclusive-NOR)
(Equivalence)
xy
x Å yx y + x y
xy
x Å y
x � yx y + x y
x y XOR
0 0 0
0 1 1
1 0 1
1 1 0
x y XNOR
0 0 1
0 1 0
1 0 0
1 1 1
Logic Operators
• NOT (Inverter)
• Buffer
x x
x x
x NOT
0 1
1 0
x Buffer
0 0
1 1
Multiple Input Gates
De Morgan’s Theorem on Gates
• AND Gate
– F = x • y F = (x • y) F = x + y
• OR Gate
– F = x + y F = (x + y) F = x • y
Change the “Shape” and “bubble” all lines
Reference
Reference Book
• Computer Organization & Architecture 7e By
Stallings
• Computer System Architecture By Mano
• Digital Logic & Computer Design By Mano
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